Shell Boundary Conditions Research Articles

Free vibration characteristics of cord-reinforced air spring with winding formation under preload conditions

A precise transfer matrix was employed to analyze the free vibration characteristics of a cord-reinforced air spring with winding formation under preload conditions. The air spring was abstracted as a rotary combined shell structure with variable winding trajectory characteristics. The rotary structure, which consisted of cylindrical and toroidal shells, was analyzed for its shell pre-stress under preload conditions. The shell pre-stress was combined with the theory of thin shells to create the rotary shell vibration control equations under preload effect. The Helmholtz equation was used to analyze the influence of sound pressure inside shells under the fluid–structure coupling boundary conditions. The rotary shell transfer matrix was derived from the variable winding trajectory equation, geometrical equation, and physical equation of shells. Under the continuous conditions of state vector between the cylindrical and toroidal shells, the shells were combined into a rotary combined shell structure to obtain its total free vibration equation. The shell boundary conditions were considered to solve the natural frequency of the air spring. The calculation results were compared with the prototype test and simulation results to verify the effectiveness and accuracy of the theoretical model. On this basis, we explored the influence of axial half wave number, circumferential wave number, geometrical structure, and material characteristics on the free vibration characteristics of the air spring. The research findings would provide important and significant guidance for the structural design of cord-reinforced air springs with winding formation.

Self-adjointness of two-dimensional Dirac operators on corner domains

We investigate the self-adjointness of the two-dimensional Dirac operator D , with quantum - dot and Lorentz - scalar \delta - shell boundary conditions, on piecewise C^2 domains (with finitely many corners). For both models, we prove the existence of a unique self-adjoint realization whose domain is included in the Sobolev space H^{1/2} , the formal form domain of the free Dirac operator. The main part of our paper consists of a description of the domain of the adjoint operator D^* in terms of the domain of D and the set of harmonic functions that verify some mixed boundary conditions. Then, we give a detailed study of the problem on an infinite sector, where explicit computations can be made: we find the self-adjoint extensions for this case. The result is then translated to general domains by a coordinate transformation.

Dynamic analysis of composite laminated doubly-curved revolution shell based on higher order shear deformation theory

The main content of this paper is to establish an analysis model for dynamic analysis of composite laminated doubly-curved revolution shell based on the Higher order Shear Deformation Theory (HSDT). Firstly, the energy functional of shell is established based on higher order shear theory. Then, the multi-segment partitioning technique is introduced to segment the shell along the generatrix direction, which is mainly to relax the boundary conditions of shell, and then reduce requirements for the displacement function. At the boundary position, the boundary spring parameters are used to obtain the corresponding boundary conditions. Similarly, the connecting spring parameters are introduced to simulate the continuity conditions between the segmented shells. The parameters of boundary spring and connecting spring can be regarded as the weight parameter. Thirdly, all displacement components of the composite laminated doubly-curved revolution shell are expressed by Jacobi orthogonal polynomials. Finally, the whole dynamic characteristics are obtained by Rayleigh-Ritz method with respect to unknown Jacobian expansion coefficient. The convergence, validity and dynamic characteristics of the analytical model established in this paper are given by a series of numerical examples.

Semi-Analytical Approach in Buckling Analysis of Functionally Graded Shells of Revolution Subjected to Displacement Dependent Pressure

Linearized buckling analysis of functionally graded shells of revolution subjected to displacement-dependent pressure, which remains normal to the shell's middle surface throughout the deformation process, is described in this work. Material properties are assumed to be varied continuously in the thickness direction according to a simple power law distribution in terms of the volume fraction of a ceramic and a metal. The governing equations are derived based on the first-order shear deformation theory, which accounts for through the thickness shear flexibility with Sanders type of kinematic nonlinearity. Displacements and rotations in the shell's middle surface are approximated by combining polynomial functions in the meridian direction and truncated Fourier series with an appropriate number of harmonic terms in the circumferential direction. The load stiffness matrix, also known as the pressure stiffness matrix, which accounts for the variation of load direction, is derived for each strip and after assembling resulted in the global load stiffness matrix of the shell, which may be unsymmetric. The load stiffness matrix can be divided into two unsymmetric parts (i.e., load nonuniformity and unconstrained boundary effects) and a symmetric part. The main part of this research is to quantify the effects of these unsymmetries on the follower action of lateral pressure. A detailed numerical study is carried out to assess the influence of various parameters such as power law index of functionally graded material (FGM) and shell geometry interaction with load distribution, and shell boundary conditions on the follower buckling pressure reduction factor. The results indicate that, when applied individually, unconstrained boundary effect and longitudinal nonuniformity of lateral pressure have little effect on the follower buckling reduction factor, but when combined with each other and with circumferentially loading nonuniformity, intensify this effect.

Study of vibration characteristics for orthotropic circular cylindrical shells using wave propagation approach and multivariate analysis

A study of free vibration of orthotropic circular cylindrical shells is presented. The vibration control equations of shells are based on Flugge classical thin shell theory. Wave approach is used in the analysis, in which the boundary conditions of shells can be simplified according to the associated beam. The free vibration frequencies of shells can be obtained from a frequency polynomial equation of order 6. The parametric analysis of the free vibration of orthotropic cylindrical shells is investigated using a statistical method. The effects of geometrical parameters and material characteristics upon frequencies are investigated here. Multivariate analysis (MVA) can be a useful tool for this parametric study. Some statistical characteristics, including correlation analysis and ANOVA are applied. ANOVA has been conducted to predict the statistical significance of the various factors. Calculations are performed in the Minitab statistical software. The results show that the L/R, h/R and m have larger effects on the lowest frequency. The importance of input parameters is ranked according to their contributions to the total variance. A knowledge and data visualization approach, Self-organizing mapping (SOM) is also adopted here for mining some intrinsic characteristics of shells.

Plastic deformations in thin rotational shells

The paper focuses on the method of calculation of evolution shells beyond the elastic limit. The conclusion of the basic system of differential equations is based on the linear shell theory with regard to the Hirchhoff-Lave hypothesis and on the physical equations for small elastic-plastic deformation theory using the method of elastic decisions. The boundary conditions are formulated for Cauchy problem: rigid attachment, hinged support, and free margin. The spherical shell boundary conditions in the pole are obtained from the conditions of symmetry and antisymmetry functions. The convergence of the elastic method and the method of the occurrence of superficial plastic deformations are studied. Also the stress-strain state in the spherical shell is determined and the convergence of the obtained solutions was studied. The results are presented on the symmetric load ring applied to the middle of the Meridian and on the load that can be considered as a concentrated force. The sufficient quantity of iterations is established to achieve the accuracy of 0.1%. The graphs are presented for radial displacement and for meridional bending moment as the functions that converge more rapidly and more slowly respectively.

Dynamic instability of parabolic shells in supersonic gas stream

The dynamic instability of the parabolic shells in a supersonic gas flow is analyzed numerically. Such structures describe a rocket fairing. Finite degrees of freedom dynamical system is obtained by the assumed-modes method to describe the shell motions. The numerical approach for calculations the modes of the shell self-sustained vibrations is suggested. The Mach numbers range of the shell dynamic instability is analyzed numerically. The influence of the paraboloid height on the regions of the dynamic instability is analyzed. The frequencies of the self-sustained vibrations are investigated. The effect of the shell boundary conditions on the regions of the dynamic instability is analyzed.

On the boundary conditions of the geometrically nonlinear Kirchhoff–Love shell theory

The present paper addresses the problem of establishing the boundary conditions of a geometrically nonlinear thin shell model, especially the kinematic ones. Our model is consistently derived from general 3D continuum mechanics statements. Generalized cross-sectional strains and stresses are based on the deformation gradient and the first Piola–Kirchhoff stress tensor. Since only the bending deformation is included in this model, no special technique needs to be adopted in order to avoid shear-locking. The theory is derived in such a way that any material model can be considered as a constitutive relation, once the zero transverse normal stress assumption is properly taken into account.Special attention is given to the question of devising the appropriate shell boundary conditions. Several parameters are proposed to characterize the boundary rotation for an arbitrary spatial shell configuration. The appearance of corner concentrated forces, related to jumps of torsion moments, is captured and justified.A weak form of the equilibrium is presented, which is suitable for implementation by means of any numerical technique that provides C1 continuity approximation. It is prone to be used with both interpolative, like the Finite Element Method, and non-interpolative methods, like the Element-Free Galerkin Method. The latter is used to exemplify the proposed approach.

EFFECT OF GEOMETRIC NONLINEARITIES ON NONLINEAR VIBRATIONS OF CLOSED CYLINDRICAL SHELLS

Geometrically nonlinear free vibrations of closed isotropic cylindrical shells are investigated through an analytical–numerical model. The method developed is a combination of Sanders–Koiter nonlinear shell theory and the finite element method. The cylindrical shell is subdivided into cylindrical finite elements and the displacement functions are derived from exact solutions of Sander's equations for thin cylindrical shells. Expressions for the mass, linear and nonlinear stiffness matrices are determined by exact analytical integration. Various boundary conditions of shell and in-plane effects are considered. Nonlinear responses are analyzed using the Runge–Kutta numerical method. The nonlinear frequency ratio is determined with respect to the amplitude thickness ratio of the motion for different study cases. Detailed numerical results are presented for various parameters for a closed isotropic shell, indicating either hardening or softening types of nonlinear behaviors, depending on the structure data. The present results show good agreement with the published ones for several cases of shells. This research clarifies the current disagreement about various types of cylindrical shells with geometric nonlinearities.

Optimal thickness distributions of aeroelastic flapping shells

The severe weight limitations of flapping wing micro air vehicles necessitates the use of thin flexible wings, which in turn requires an aeroelastic modeling tool for proper numerical characterization. Furthermore, due to the unconventional nature of these vehicles, wing design guidelines for thrust and/or power considerations are not generally available; numerical design optimization then becomes a valuable tool. This work couples a nonlinear shell model to an unsteady vortex lattice solver, and then computes analytical design gradients: the derivative of aerodynamic force/power quantities with respect to a large vector of thickness variables. Gradient-based optimization is then used to locate the wing structure that maximizes the thrust, or minimizes the power under a thrust constraint, for a variety of shell boundary conditions. Changes in the topological features of the optimal wing thicknesses highlight important aeroelastic interactions that can be exploited for efficient flapping wings.

Local adaptive differential quadrature for free vibration analysis of cylindrical shells with various boundary conditions

This paper presents the formulation and numerical analysis of circular cylindrical shells by the local adaptive differential quadrature method (LaDQM), which employs both localized interpolating basis functions and exterior grid points for boundary treatments. The governing equations of motion are formulated using the Goldenveizer–Novozhilov shell theory. Appropriate management of exterior grid points is presented to couple the discretized boundary conditions with the governing differential equations instead of using the interior points. The use of compactly supported interpolating basis functions leads to banded and well-conditioned matrices, and thus, enables large-scale computations. The treatment of boundary conditions with exterior grid points avoids spurious eigenvalues. Detailed formulations are presented for the treatment of various shell boundary conditions. Convergence and comparison studies against existing solutions in the literature are carried out to examine the efficiency and reliability of the present approach. It is found that accurate natural frequencies can be obtained by using a small number of grid points with exterior points to accommodate the boundary conditions.

ON THE FREE VIBRATION OF STIFFENED SHALLOW SHELLS

A finite element analysis for free vibration behaviour of doubly curved stiffened shallow shells is presented. The stiffened shell element is obtained by the appropriate combinations of the eight-/nine-node doubly curved isoparametric thin shallow shell element with the three-node curved isoparametric beam element. The shell types examined are the elliptic and hyperbolic paraboloids, the hypar and the conoidal shells. The accuracy of the formulation is established by comparing some of the authors' results of specific problems with those available in the literature. Numerical results of additional stiffened shells are also presented to study the effects of various parameters of shells and stiffeners such as orientation (i.e., along x -/ y -/both x and y directions), type (concentric, eccentric at top and eccentric at bottom) and number of stiffeners, stiffener depth to shell thickness ratio, and aspect ratio, shallowness and boundary conditions of shells on free vibration characteristics.

Luttinger-liquid state of the zigzag double chain

The zigzag double chain is investigated using Lanczos techniques. The weak-coupling limit is similar to that of a ladder system which is known to have a spin gap at half-filling at all coupling strengths. But the half-filled zigzag system is gapless at strong coupling. The question whether a spin gap occurs upon doping is investigated. It is shown that the doped zigzag chain is a Luttinger liquid in which the holes repel each other. The criterion is based on exact diagonalization techniques for open shell boundary conditions. As a further corroboration, the hole-hole and the hole-spin correlations are also computed.

Stability and Teller’s theorem: Fullerenes in the March model

We study C${}_{60}$ with the use of the March model [N. H. March, Proc. Camb. Philos. Soc. 48, 665 (1952)]. A spherical shell model is invoked to treat the nuclear potential, where the nuclear and core charges are smeared out into a shell of constant surface charge density. The valence electron distribution and the electrostatic potential are efficiently computed by integration of the Thomas-Fermi equation, subject to the shell boundary conditions. Total energy is numerically calculated over a range of shell radii, and the mechanical stability of the model is explored with attention to the constraints of Teller's theorem [E. Teller, Rev. Mod. Phys. 34, 627 (1962)]. The calculated equilibrium radius of the shell is in fair agreement with experiment.

両端に回転ばねのある部材で構成される単層ラチスドームの線形および弾性座屈荷重 : 矩形平面形状をした裁断球殼状ドームについて

The present study discusses the elastic buckling load of spherical latticed domes on rectangular plan. The buckling loads are non-dimensionalized by using both the buckling loads as pinconnected dome with no bending rigidity at the ends of members and the buckling load as an equivalent. shell. The dividing parameter by which the dome can be classified like a pinconnected dome or like a continuous shell is defined quantitatively and is proved to depend on the boundary conditions of shells. Also the elastic buckling loads are shown to fall on nearly half of their linear buckling loads obtained by neglecting the prebuckling deformations.

Generalized expressions for boundary conditions of shells of revolution

For the boundary conditions of shells of revolution, traditionally, four out of the eight quantities which are the four displacements on the middle surface u, v, w and ϕ together with the four corresponding forces, are given. When the generalized displacements on the nodal circles are used as basic unknowns, the number of unknowns on a nodal circle is more than four[1][2][3][4]. In this case, how to deal with the boundary conditions is still a problem that has not been solved satisfactorily yet. In this paper, the relations between the generalized and nongeneralized quantities of a shell's edge are derived according to the principle of virtual work. Seven types of common edges are studied and their expressions of boundary conditions in the form of generalized displacements or forces are given. The number of expressions for each type of edge may correspond with the number of unknowns used on a nodal circle. With these expressions, boundary conditions can be put directly into equations of motion of generalized displacement method so as to solve the generalized displacements. By so doing, the process of transformation and inverse transformation about unknowns in [2] is avoided. Not only is the argument simple and clear, but the calculation work is reduced. Having the set of generalized expressions of boundary conditions, the generalized displacement method of the shell of revolution may be more perfect in theory.

Transient Motion of an Elastic Shell of Revolution in an Acoustic Medium

A numerical solution for the transient vibration of an arbitrary shell of revolution, surrounded by an acoustic medium, has been formulated in terms of the conformally mapped acoustic equations, fluid shell boundary conditions and spectral representation of the shell displacements. The technique readily includes the boundary condition at infinity, internal shell structure and is exact to within those approximations implicit in the finite-difference method.

Vibration and Buckling of Anisotropic Shallow Shells

The free vibration frequencies and the critical buckling loads of certain thinwalled shells of revolution constructed of anisotropic materials—layered or heterogeneous—are studied, the specific geometries being circular cylinders, barrels of positive and negative Gaussian curvature, and spherical sections. The buckling configurations include axial compression and simple radial pressure. In each case the shell boundary conditions are simple supports. The analyses are based upon shallow shell theory, linearized about an initial membrane pressurization state. Using known modal techniques, a frequency determinant is derived. From this determinant studies of the effects of in-plane inertia, of various wave numbers, and of the physical parameters of the shell are carried out. Additionally, results for the static buckling of these shells under axial compression and radial pressure are also obtained, using the kinetic buckling criterion. Parameters studied in the vibration and stability analyses include the thickness-to-radius ratio, initial pressurization, circumferential and axial wave numbers, and the degree of anisotropic asymmetry. The latter effect is studied by the comparison of analyses of two different layered anisotropic materials—each constructed of five equa thickness orthotropic layers with varying orientations. The orientations have been selected to produce varieties of anisotropy—midplane symmetric and midplane asymmetric. The results of these free vibration and stability analyses have produced the following conclusions: (1) the effect of anisotropic asymmetry is so significant that it must be considered; (2) the increased stiffening found for orthotropic cylinders as a consequence of increased thickness ratio and initial pressurization also was found for the general anisotropic family of shells studied.

Effective boundary conditions for reflection and transmission by an absorbing shell of arbitrary shape

The reflection and transmission properties of an absorbing shell of arbitrary shape can be described by two algebraic boundary conditions relating the four tangential fields E_{+}, H_{+}, E_{-}, H_{-} at corresponding points on the outer surface S_{+} and inner surface S_{-} , respectively. The boundary conditions contain first-order curvature correction terms, but it is still necessary that the shell curvature be reasonably small compared to the attenuation per unit distance in the shell material. Especially simple conditions are obtained for shells thin compared to the attenuation length. The shell boundary conditions can also be used to determine an effective surface impedance for a conducting body coated by multiple layers of absorbing material.